DEPARTMENT OF MATHEMATICS - Caedmon … resources and links to video tutorials; exam papers and solutions, formula booklet: Use the Mathematics department website: ccwmaths.wordpress.com

DEPARTMENT OF MATHEMATICS

A2 level Mathematics

Core 3 course workbook

2015-2016

Name: _____________________________________

Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for revision and that it will help you enjoy and benefit from the course more. ORGANISATION Keep this workbook, worksheets, homework and assessments in a file + BOOK SCRUTINY You will be asked to hand this in/bring it along to a progress interview with

your teacher or Mr Gower during the year REVISION BOOKLET We will produce revision a revision programme once you have completed TRIAL EXAM all or most of the Core 3 course. As last year there will be an expectation that

you do one chapter a week/fortnight to prepare for trial and actual exams. There will be one opportunity to resit the trial exam

You may also find the following useful: Free Electronic copy of C3 textbook: (linked from the department website)

Dropbox.com For resources and links to video tutorials; exam papers and solutions, formula booklet: Use the Mathematics department website:

ccwmaths.wordpress.com

And CCW Maths on twitter: @ccwmaths

For exam practice questions with solutions or exam past papers use (linked from the department website)

Examsolutions.net Physicsandmathstutor.com

Mathspapers.co.uk

Mr J Gower Head of KS5 Mathematics

FORMULA SHEET

Exponential and natural logarithm functions:

In general: ex = y and so: x = ln y

The usual laws of logarithms apply, so:

ln a + ln b = ln ab ln a – ln b = ln

b

a ln ak = k ln a

Trigonometric Identities

You need to know the trig identities from C2:

tan

cos

sin and 1cossin 22

You can easily use the second of these to derive these C3 identities:

xx 22 tan1sec and cosec 2 x x2cot1

There are a few other identities which you need to be able to use (but the next few are given in the

formula book):

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

BA

BABA

tantan1

tantan)tan(

By making A = B in the above identities, you can easily derive these double angle formulae (these

are not given in the formula book):

sin 2A = 2sin A cos A

cos 2A = cos2A – sin2A = 2 cos2 𝑥 − 1 = 1 − 2 sin2 𝑥

A

AA

2tan1

tan22tan

Reciprocal Trig functions

𝑐𝑜𝑠𝑒𝑐 𝑥 = 1

sin 𝑥, sec 𝑥 =

1

cos 𝑥, cot 𝑥 =

1

tan 𝑥,

Inverse Trig functions

𝑎𝑟𝑐𝑠𝑖𝑛 𝑥 = sin−1 𝑥 , 𝑎𝑟𝑐𝑐𝑜𝑠 𝑥 = cos−1 𝑥 , 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 = tan−1 𝑥,

Differentiation:

The Chain Rule (for differentiating a function of a function)

Very simply: dx

du

du

dy

dx

dy (where y is a function of u, and u is a function of x)

The Product Rule (for differentiating products of functions)

If u and v are functions of x, and y = uv, then:

dx

duv

dx

dvu

dx

dy (this is not given in the formula book, so learn it!)

The Quotient Rule (for differentiating quotients of functions)

If u and v are functions of x, and y =v

u, then:

2v

dx

dvu

dx

duv

dx

dy

(this is in the formula book, but in a different form)

Trig differentiation

𝑓(𝑥) 𝑓′(𝑥)

sin 𝑘𝑥 k cos 𝑘𝑥

cos 𝑘𝑥 − k sin 𝑘𝑥

tan 𝑘𝑥 k sec2 𝑘𝑥

cosec 𝑥 − cosec 𝑥 cot 𝑥

sec 𝑥 sec 𝑥 tan 𝑥

cot 𝑥 −cosec2 𝑥

Contents

1. Algebra and Functions

2. Exponential and logarithm functions

3. Trigonometry

4. Differentiation

5. Numerical methods

6. Revision

C3WB: Algebra and Functions

page

Algebra Notes • BAT find equivalent expressions by dividing by the denominator

• BAT add, subtract, multiply and divide rational expressions

• BAT simplify rational expressions


page

WB1 Express 𝑥

(𝑥+3)−

6

(𝑥−1) as a single fraction in its simplest form.

WB2

Common

factor in the

denominator

Express ...

2

3

4

32

xx as a single fraction in its simplest form


page

WB3

Exam Q Express

4𝑥−4

2(𝑥−4)−

36

2(𝑥−4)(2𝑥−5) as a single fraction in its simplest form

Given that

𝑓(𝑥) = 4𝑥 − 4

2(𝑥 − 4)−

36

2(𝑥 − 4)(2𝑥 − 5)− 2

b) Show that

𝑓(𝑥) = 12

(2𝑥 − 5)


page

WB4

Simplify: 12𝑥−96

𝑥(𝑥−8)(7𝑥+7)÷

3

(7𝑥+7)2


page

Top heavy fractions Notes

BAT Manipulate algebra fluently and efficiently

BAT Simplify top heavy algebraic fractions


page

WB5

Top heavy

fraction

Simplify 1

225 2

x

xx

WB6

Top heavy

fraction (evil)

Show that 4127

3251422

23

x

CBAx

xx

xxx


page

Functions Introduction notes

BAT understand definition of a function as a one to one mapping

BAT recognise odd and even functions

BAT state the domain and range of functions


page

WB7 Sketch the graph of y = f(x) where f(x) = x2 + 1, x ≥ -2

What is the domain of f ?

What is the range of f ?


page

WB8 The function h(x) is defined by ℎ(𝑥) = 1

𝑥 + 2, 𝑥𝜖ℝ 𝑥 ≠ 0

Solve these equations

h(x) = 3 h(x) = 4 h(x) = 1

Explain why the equation h(x) = 2 has no solution

Sketch a graph of h(x)


page

WB9 Draw a sketch of the function defined by:

𝑓: 𝑥 ⟼ (2𝑥 + 1, − 3 < 𝑥 < 4

13 − 𝑥, 4 ≤ 𝑥 < 10

)

and state the range of f(x)


page

WB10 Draw a sketch of the function defined by g(t) = 3t + 2,

and state it’s domain and range


page

WB11 Draw a sketch of the function defined by 𝑓(𝑥) = 6

𝑥+1, x 1

State it’s domain and range

Is f(x) an odd function? Give a reason for your answer


page

Inverse Functions notes

BAT Find the inverse of functions

BAT Find the domain and range of inverse functions

BAT understand the inverse of a function graphically


page

WB12 Find the Inverse function: x

xxf ,5

3)(

2

WB13

Find the Inverse function: 𝑓(𝑥) = 6

3𝑥−2, 𝑥 ≠

2

3


page

WB14 Find the Inverse function: 𝑓(𝑥) =

3𝑥+2

2𝑥−5

WB15

Find the Inverse function:

xx

xf ,53

12sin)(


page

WB16

Inverse function graphically

Sketch on the same axes f(x) and its inverse

Sketch on the same axes g(x) and its inverse


page

Composite Functions notes

BAT Find composite functions

BAT Find the domain and range of composite functions


page

WB 17 A gas meter indicates the amount of gas in cubic feet used by a consumer. The

number of therms of heat from x cubic feet of gas is given by the function f

where f(x) = 1.034x, x > 0

A particular gas company’s charge in £ for t therms is given by the function g

where g(t) = 15 + 0.4t

(i) Find the cost of using 100 cubic feet

(ii) Find a single rule for working out the cost given the number of cubic

feet of gas used

WB 18

The functions f and g are defined by: 12:

,: 2

xxg

xxxf

Find i) fg(x) ii) fg(0) iii) gf(x) iv) gf(1) v) ff(x) vi) ff(-2)

vii) gg(x) viii) gg(-7)


page

WB 19 The function f has domain [−∞, ∞] and is defined by 𝑓(𝑥) = 4𝑒𝑥

The function g has domain [1, ∞] and is defined by g(𝑥) = 3 ln 𝑥

a) Explain why 𝑔𝑓(−3) does not exist

b) Find in its simplest form an expression for 𝑓𝑔(𝑥) stating its domain and

range


page

Modulus Functions and transformations notes

BAT Understand and draw graphs of modulus functions

BAT Find intersections of graphs, including modulus graphs

BAT Solve inequalities – three methods

BAT apply transformations to graphs


page

Review. Transformations of graphs

Learn these and practice

them

f(x) = x3

sketch f(3x)

sketch f(x+2)

sketch 4 f(x)

f(x) = 2x2 + 3

sketch f(3x)

sketch f(x+2)

sketch 4 f(x)

f(x) = 2x3 + 4x

sketch f(2x)

sketch f(x – 1)

sketch 3f(x)

f(x) = Sin x

sketch f(2x)

sketch f(x + 90)

sketch 3 f(x)

f(x) = Cos x

sketch – f(x)

sketch f(x – 30)

sketch 2 f(x)

f(x) = Tan x

sketch f(2x)

sketch f(x + 180)

sketch f (– x)

(i) Shifts

(ii) Stretches

(iii) Reflections

+A

(x, y) (x + A, y)

+A

(x, y)

(x , y + A)

×

(x, y)

( x , y)

×A

(x, y)

(x, Ay)

(x, – y)

(x, y)

(x, y) (-x, y)

Axf )( is a shift

in the y direction

)( Axf is a shift

in the x direction

)(Axf is a stretch by scale

factor A

1 in the x direction

)(xAf is a stretch by scale

factor A in the y direction

)( xf is a reflection of

the graph in the y axis

)(xf is a reflection of

the graph in the x axis


page

WB 20 The functions f and g are defined by: 𝑓(𝑥) = 𝑥 + 3

𝑔(𝑥) = |4𝑥 + 6| Solve the inequality 𝑓𝑔(𝑥) > 12


page

WB21

Solve the Inequality |𝑥 − 12| > |3𝑥|


page

WB22 Given that 𝑓(𝑥) = |𝑥| and g(𝑥) = 𝑥 + 3

Sketch the graphs of the composite functions 𝑓𝑔(𝑥) 𝑎𝑛𝑑 𝑔𝑓(𝑥)

Indicating clearly which is which


page

WB23

Solve the inequality |3𝑥 − 1| < |6𝑥 − 1|


page

WB24

a) Solve the equation |9𝑥2 − 61| = 60

b) Hence, or otherwise, solve the inequality |9𝑥2 − 61| ≥ 60


page

WB25 𝑓(𝑥) = 𝑥4 − 4𝑥 − 240

a) Show that there is a root of f(x) = 0 in the interval [-4, -3]

b) Find the coordinates of the turning point on graph of y = f(x)

Given that 𝑓(𝑥) = (𝑥 − 4)(𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐)

find the values of a,b and c

d) Sketch the graph of y = f(x)

e) Hence sketch the graph of y = f(x)

C3 WB: Exponential & natural Logarithm

page

Exponential and Natural logarithm Graphs - Notes BAT Understand and sketch transformations of exponential and natural

logarithm graphs


page

WB1

Sketch the graph of: y = 10e-x

WB2

Sketch the graph of: y = 3 + 4e0.5x


page

WB3

Sketch the graph of: y = 3 + ln(2x)

WB4

Sketch the graph of: y = ln(3 - x)


page

Exp and ln - Solving equations - Notes BAT solve equations and manipulate algebra using rules of indices and

logarithms


page

WB5 Solve

i) 𝑒𝑥 = 3

ii) 𝑒𝑥+2 = 7

iii) ln 𝑥 = 4

iv) ln(3𝑥 − 2) = 3

WB6 Make x the subject of:

i) 𝑙𝑛𝑦 − 𝑙𝑛𝑥 = 4𝑡

ii) 𝑦 = 5𝑒2𝑥


page

WB7 Solve

i) 𝑙𝑛𝑥 −16

𝑙𝑛𝑥= 6

ii) 𝑒2𝑥 − 8𝑒𝑥 + 12 = 0

WB8

Exam Q

Find the exact solutions to these equations

a) ln(3𝑥 − 8) = 2

b) 3𝑥𝑒8𝑥+3 = 18


page

Growth Functions - Notes BAT Solve real life problems involving growth functions of the

form y = Aebx+c


page

WB9 The Price of a used car is given by the formula: 𝑃 = 1600𝑒−

𝑡

10

a) Calculate the value of the car when it is new

b) Calculate the value after 5 years

c) What is the implied value of the car in the long run (ie – what value does it

tend towards?)

d) Sketch the Graph of P against t


page

WB10 The number of elephants in a herd can be represented by the equation: 𝑁 = 150 − 80𝑒−

1

40

Where n is the number of elephants and t is the time in years after 2003.

a) Calculate the number of elephants in the herd in 2003

b) Calculate the number of elephants in the herd in 2007

c) Calculate the year when the population will first exceed 100 elephants

d) What is the implied maximum number in the herd?

C3WB: Trigonometry

page

Trig Reciprocal Functions - Notes BAT understand sec, cosec and cot and their graphs

BAT solve equations involving sec, cosec and cot

C3WB: Trigonometry

page

WB1

a) Will cosec 200 be positive or negative?

b) Find the value of sec 280 to 2 decimal places

WB2 Solve these equations: ( 0 < 2π )

a) 𝒔𝒆𝒄 𝜽 = 𝟏𝟎

b) 𝟓 𝒄𝒐𝒔𝒆𝒄 𝜽 = 𝟐𝟓

c) 𝒄𝒐𝒕 𝟑𝜽 + 𝟕 = 𝟔. 𝟓

C3WB: Trigonometry

page

Graphs of Reciprocal functions - sec x, cosec x, cot x

C3WB: Trigonometry

page

WB3 Sketch, in the interval 0 ≤ θ ≤ 360,

the graph of 𝑦 = 1 + sec 2𝜃

C3WB: Trigonometry

page

Trig Identities and Equations - Notes BAT prove identities using reciprocal functions

BAT know and use new trig identities

C3WB: Trigonometry

page

WB4

Simplify 𝑠𝑖𝑛𝜃 𝑐𝑜𝑡𝜃 𝑠𝑒𝑐 𝜃

WB5

Simplify sin 𝜃 cos 𝜃 (sec 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃)

C3WB: Trigonometry

page

WB6

Show that cot𝜃 𝑐𝑜𝑠𝑒𝑐 𝜃

𝑠𝑒𝑐2𝜃+ 𝑐𝑜𝑠𝑒𝑐2𝜃≡ 𝑐𝑜𝑠3𝜃

WB7 Prove that 𝑐𝑜𝑠𝑒𝑐4𝜃 − 𝑐𝑜𝑡4𝜃 ≡

1+𝑐𝑜𝑠2𝜃

1−𝑐𝑜𝑠2𝜃

C3WB: Trigonometry

page

WB8

Prove that 𝑠𝑒𝑐2𝜃 − 𝑐𝑜𝑠2𝜃 ≡ 𝑠𝑖𝑛2𝜃 (1 + 𝑠𝑒𝑐2𝜃)

WB9

Solve the equation 4𝑐𝑜𝑠𝑒𝑐 𝜃 − 9 = cot 𝜃 in the interval 0 ≤𝜃 ≤ 360

C3WB: Trigonometry

page

WB10

Solve these equations:

a) 4𝑐𝑜𝑠2𝜃 + 5 sin 𝜃 = 3 in the interval −𝜋 ≤ 𝜃 ≤ 𝜋

b) 2𝑡𝑎𝑛2𝜃 + sec 𝜃 = 1 in the interval −𝜋 ≤ 𝜃 ≤ 𝜋

C3WB: Trigonometry

page

Inverse trig questions - Notes BAT use and understand applications and graphs of the inverse Trig functions

C3WB: Trigonometry

page

Graphs of arcsin x, arcos x, arctan x

C3WB: Trigonometry

page

WB11

a) Work out in degrees, the value of arcsin (−√2

2)

b) Work out in radians, the value of cos [arcsin (−1)]

WB12

a) Work out, in radians, the value of arcsin(0.5)

b) Work out, in radians, the value of arctan(√3)

C3WB: Trigonometry

page

WB13

a) Work out, in radians, the value of arcsin(0.5)

b) Work out, in radians, the value of arctan(√3)

C3WB: Trigonometry

page

Addition Formulae - Notes BAT understand where the addition formula come from and know the Identities for

sin 2x, cos 2x and tan 2x

BAT use the addition formulae to solve ‘show that’ problems

BAT use the addition formulae to solve equations

C3WB: Trigonometry

page

WB14

Show that sin 15 =√6 − √2

4

WB15 Given that sin 𝐴 = −

3

5 in the range 180 < 𝐴 < 270 and cos𝐵 =

− 12

13 where B is Obtuse . Find the value of tan(𝐴 + 𝐵)

C3WB: Trigonometry

page

WB16

Rewrite the following as a single Trigonometric function: 2 sin𝜃

2cos

𝜃

2cos 𝜃

WB17

Show that 1 + cos 4𝜃 can be written as 2𝑐𝑜𝑠22𝜃

C3WB: Trigonometry

page

WB18

Given that cos 𝑥 =3

4 in the range [180, 360] find the exact value of 2 sin 2𝑥

WB19 Given that x = 3 sin 𝜃 and 𝑦 = 3 − 4 cos 2𝜃

Eliminate and express y in terms of x

C3WB: Trigonometry

page

WB20

Solve the equation 3 cos 2𝑥 − 𝑐𝑜𝑠 𝑥 + 2 = 0 in the range 0 ≤ 𝑥 ≤ 360

C3WB: Trigonometry

page

R cos ( x + alpha) - Notes BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a

sine or cosine function only

BAT solve equations of the form acosθ + bsinθ, or find their max/min value

C3WB: Trigonometry

page

WB21

Show that 3 sin 𝑥 + 4 cos 𝑥 can be expressed in the form 𝑅 sin(𝑥+∝)

where 𝑅 > 0, 0 < 𝛼 < 90

WB22

a) Show that you can express sin 𝑥 − √3 cos 𝑥 in the form 𝑅 sin(𝑥+∝)

where 𝑅 > 0, 0 < 𝛼 < 90

b) Hence, sketch the graph of y = sin 𝑥 − √3 cos 𝑥

C3WB: Trigonometry

page

WB23

a) Express 2 cos 𝜃 + 5 sin 𝜃 can be expressed in the form 𝑅 cos(𝜃−∝)

where 𝑅 > 0, 0 < 𝛼 < 90

b) Hence, solve 2 cos 𝜃 + 5 sin 𝜃 = 3 in the range 0 < 𝜃 < 360

WB24

a) Express 12 cos 𝜃 + 5 sin 𝜃 in the form 𝑅 cos(𝜃−∝)

where 𝑅 > 0, 0 < 𝛼 < 90

b) Hence, find the maximum value of 12 cos 𝜃 + 5 sin 𝜃 and the smallest positive

value of at which it arises

C3WB: Trigonometry

page

WB23

a) Express 2 cos 𝜃 + 5 sin 𝜃 can be expressed in the form 𝑅 cos(𝜃−∝)

where 𝑅 > 0, 0 < 𝛼 < 90

b) Hence, solve 2 cos 𝜃 + 5 sin 𝜃 = 3 in the range 0 < 𝜃 < 360

WB24

a) Express 12 cos 𝜃 + 5 sin 𝜃 in the form 𝑅 cos(𝜃−∝)

where 𝑅 > 0, 0 < 𝛼 < 90

b) Hence, find the maximum value of 12 cos 𝜃 + 5 sin 𝜃 and the smallest positive

value of at which it arises

C3WB: Differentiation

page

Chain rule - Notes BAT differentiate a function of a function, using the chain rule

BAT apply 𝒅𝒚

𝒅𝒙=

𝟏𝒅𝒙

𝒅𝒚

to solve a problem


page

WB1

Differentiate 𝑦 = (3𝑥4 + 𝑥)5

WB2

Given that 𝑦 = √5𝑥2 + 1 Find the value of f’(x) at (4,9)


page

WB3 Given that 𝑦 = (𝑥2 − 7𝑥)4 Calculate

𝑑𝑦

𝑑𝑥 using the chain rule

WB4 Given that 𝑦 =

1

√6𝑥−3

Calculate 𝑑𝑦

𝑑𝑥 using the chain rule


page

WB5 Calculate 𝑑𝑦

𝑑𝑥 for the following equation 𝑦3 + 𝑦 = 𝑥

and find the gradient at the point (2,1)


page

Product rule - Notes BAT differentiate a product of two functions


page

WB6

differentiate 𝑦 = 𝑥2(3𝑥 − 9)

WB7

Differentiate 𝑦 = (𝑥2 + 3)(4𝑥 + 1)


page

WB8

Given that: 𝑓(𝑥) = 𝑥(2𝑥 + 1)3, find f’(x)

WB9

Given that: 𝑓(𝑥) = 𝑥2√3𝑥 − 1, find f’(x)


page

Chain, Product and Quotient rules - Notes BAT differentiate a quotient of two functions

BAT differentiate complex functions using the chain rule, product rule

and quotient rules efficiently


page

WB10

Given that 𝑦 =𝑥

2𝑥+5 Calculate

𝑑𝑦

𝑑𝑥

WB11

Given that 𝑦 =3𝑥

(𝑥+4)2 Calculate

𝑑𝑦

𝑑𝑥


page

Exponential and natural logarithm functions- Notes BAT differentiate the exponential and natural logarithm functions

BAT apply the chain. Product and Quotient rules

BAT solve geometry problems using what we have learned so far


page

WB12

Chain rule

a) Find the gradient of 𝑦 = 5 ln 𝑥 + 𝑒−𝑥 − 4 ln 2𝑥 when 𝑥 = 4

b) Find the gradient of 𝑦 = 3 ln(3 𝑥2 + 6𝑥) when 𝑥 =1

2


page

WB13

Product rule

Differentiate the following: a) 𝑦 = 𝑥 𝑒𝑥 b) 𝑦 = 𝑥3 ln 𝑥


page

WB14

Quotient rule

Differentiate the following: a) 𝑦 = 𝑒3𝑥−1

𝑥 b) 𝑦 =

𝑙𝑛5𝑥

𝑥


page

WB15

Geometry

Find the equation of the tangents to the curve 𝑦 = 3𝑒−5𝑥

at the point where 𝑥 = 0

WB16

Geometry

Find the equation of the normal to the curve 𝑦 = −1

4ln 𝑥

at the point where 𝑥 = 1


page

Trig functions - Notes BAT differentiate trig functions using what we have learned so far

BAT differentiate mixed trig, exponential, logarithm and other functions


page

WB17

Differentiate: a) 𝑦 = sin 3𝑥 b) 𝑦 = sin2

3𝑥 c) 𝑦 = 𝑠𝑖𝑛2𝑥

WB18

Differentiate: a) 𝑦 = 𝑐𝑜𝑠(4𝑥 − 3) b) 𝑦 = 𝑐𝑜𝑠3𝑥 c) 𝑦 = 3 cos 𝑥


page

WB19

Find the derivative of 𝑦 = tan 𝑥

WB20

Find the derivative of 𝑦 = tan 4𝑥


page

WB21

Find the derivative of 𝑦 = 𝑥 tan 2𝑥

WB22

Find the derivative of 𝑦 =ln 𝑥

sin𝑥


page

WB23

Find the derivative of 𝑦 = 𝑒𝑥 sin 𝑥


page

Reciprocal Trig functions - Notes BAT differentiate reciprocal trig functions using what we have learned so far

BAT differentiate mixed trig, exponential, logarithm and other functions


page

WB24 Find the derivative of 𝑦 = cot 𝑥

Find the derivative of 𝑦 = sec 𝑥

Find the derivative of 𝑦 = cosec 𝑥


page

WB25

Find the derivative of 𝑦 = sec 3𝑥

WB26

Find the derivative of 𝑦 =𝑐𝑜𝑠𝑒𝑐 2𝑥

𝑥2

C3WB: Numerical methods

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Notes BAT Show that a root of an equation lies between two values

BAT Approximate solutions using an iterative formula

BAT check solutions with an appropriate method


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WB1

Show that the equation 𝑓(𝑥) = ln 𝑥 − 1 Has a root between 2 and 3

WB2 Find a root of the equation 𝑥3 − 12𝑥 + 12 = 0


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WB3

Find a root of the equation 𝑒−𝑥 − 𝑥 = 0

WB4 α is the positive root of the equation 𝑥3 −1

𝑥− 1 = 0

Show that to 3 decimal places α = 1.221


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WB5

Exam Q The curve 𝑦 = 5𝑥 intersects the line 𝑦 = 𝑥 + 2 at the point where 𝑥 = 𝛼

a) Show that α lies between 0 and 1

b) Show that the equation 5𝑥 = 𝑥 + 2 can be rearranged into x = ln(𝑥+2)

ln 5

c) Use the iteration 𝑥𝑛+1 = ln(𝑥𝑛+2)

ln 5 with 𝑥1 = 0 to find 𝑥3 to two sf

WB6

Exam Q 𝑓(𝑥) = −𝑥3 + 7𝑥2 − 5

a) Show that the equation f(x) = 0 can be rewritten as 𝑥 = √5

7−𝑥

b) Starting with 𝑥1 = 0.9 , use the iteration 𝑥𝑛+1 = √5

7−𝑥𝑛 to calculate the values

of 𝑥2, 𝑥3, and 𝑥4 giving your answers to 4 decimal places

c) Show that 𝑥 = 0.906 is a root of f(x) = 0 correct to 3 decimal places

Documents

DEPARTMENT OF MATHEMATICS - Caedmon … resources and links to video tutorials; exam papers and solutions, formula booklet: Use the Mathematics department website: ccwmaths.wordpress.com